微分系统解析近似解的符号计算研究

发布时间：2018-06-13 20:04

本文选题：微分方程 + Laplace变换　；参考：《华东师范大学》2015年硕士论文

【摘要】：非线性微分方程的解法研究是当今非线性科学的一个重要研究内容.Adomian分解法是构造非线性微分方程解析近似解的一种有效方法,该方法因为思路简单而获得了广泛应用.但因为符号计算中间表达式急剧膨胀问题,致使单纯使用Adomian分解法获得的解析近似解的收敛区间往往很有限.近期有学者将Laplace变换法和Adomian分解法相结合,即所谓的Laplace分解法Laplace分解法较已有的Adomian分解法计算效率更高.本文将Laplace分解法推广应用到非线性偏微分方程情形,并针对已有算法的缺陷,提出了改进的Laplace分解法.此外,本文还基于这两种算法研发了自动推导非线性微分系统解析近似解的软件LDM P.本文的主要内容如下：第一章主要介绍了和本文工作相关的研究背景,回顾了非线性微分系统解法研究的发展历程,并简要总结了国内外在该领域所取得的成果与发展现状.第二章主要介绍了Laplace分解法及其改进算法.首先阐述了直接推广的Laplace分解法的思路与过程,然后通过具体实例对算法的缺陷进行分析,进而提出了改进的Laplace分解法,并通过求解不同类型的方程对改进前后两种算法的适用范围、优缺点等作了比较,由此可知,Laplace分解法对扩大级数解的收敛区间、提高级数解的精度均有很好的效果.第三章主要介绍了非线性微分系统解析近似解的自动推导软件LDMP.简要介绍了软件的使用接口及其中主要模块的功能和实现思路.通过应用到不同类型的方程实例,进一步验证了算法及软件的有效性.软件LDMP界面友好,使用方便,其编写过程中局部也采用了并行化的思想和方法.用户只要按照格式要求输入待求解的方程及可能的初边值条件,LDMP即可自动输出所获得的解析近似解,还可输出不同阶解的比较曲线及误差曲线,由此可进一步佐证所获结果的有效性.
[Abstract]:The study of solving nonlinear differential equations is an important research content of nonlinear science. Adomian decomposition method is an effective method to construct analytical approximate solutions of nonlinear differential equations. This method has been widely used because of its simple thinking. However, due to the problem of sharp expansion of intermediate expressions in symbolic computation, the convergence interval of analytical approximate solutions obtained by using Adomian decomposition method is often very limited. Recently, some scholars have combined the Laplace transform method with the Adomian decomposition method, that is, the so-called Laplace decomposition method is more efficient than the existing Adomian decomposition method. In this paper, the Laplace decomposition method is extended to nonlinear partial differential equations, and an improved Laplace decomposition method is proposed to overcome the defects of the existing algorithms. In addition, based on these two algorithms, a software called LDMP is developed for the automatic derivation of analytical approximate solutions of nonlinear differential systems. The main contents of this paper are as follows: the first chapter introduces the research background related to the work of this paper, reviews the development history of the nonlinear differential system solution, and briefly summarizes the achievements and development status in this field at home and abroad. In chapter 2, the Laplace decomposition method and its improved algorithm are introduced. In this paper, the idea and process of the Laplace decomposition method, which is extended directly, is introduced, and then the defects of the algorithm are analyzed through concrete examples, and the improved Laplace decomposition method is put forward. By solving different kinds of equations, the application range, advantages and disadvantages of the two algorithms before and after the improvement are compared. It can be seen that the Laplace decomposition method has a good effect on enlarging the convergence interval of the series solution and improving the accuracy of the series solution. In chapter 3, the analytical approximate solution of nonlinear differential system is introduced. This paper briefly introduces the interface of the software, the function and realization of the main modules. The validity of the algorithm and the software is further verified by the application of different kinds of equations. The software LDMP has friendly interface and easy to use. In the process of programming, it also adopts the idea and method of parallelization. The user can automatically output the analytical approximate solution obtained by the input of the equation to be solved and the possible initial boundary value condition according to the format requirement, and can also output the comparison curve and error curve of different order solutions. This can further verify the validity of the obtained results.
【学位授予单位】：华东师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175

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